This book constitutes the thesis of the author at the University of Mainz.
The mathematical objects studied by the author are manifolds of bounded geometry defined as follows:
“Let \(M\) be a manifolds with boundary \(\partial M\). Then \(M\) is of bounded geometry if:
(N) There exists \(r_C>0\) so that the geodesic collar \(e:[0,r_C) \times \partial M\to M:(t,x)\to \exp_x(tv_x)\) is a diffeomorphism onto its image \((v_x\) is the unit inward normal vector at \(x\in \partial M)\).
(IC) The injectivity radius of \(\partial M\) is positive.
(I) There is \(r_i>0\) so that for \(x\in M-e ([0,{r_C \over 3}) \times \partial M)\) the exponential map is a diffeomorphism on \(B(0,r_i) \subset T_xM\).
(B) For every \(K\in\mathbb{N}\) there is \(C_K>0\) so that \(| \nabla^i R|\leq C_K\) and \(|\overline \nabla^i l|\leq C_K\), \(0\leq i\leq K\) \((R=\)curvature, \(l=\)second fundamental form tensor, \(\nabla=\)Levi-Civita connection of \(M\) and \(\overline \nabla=\)Levi-Civita connection of \(\partial M)\).”
The author establishes several interesting results about the geometry, topology and analysis on this type of manifolds, directed at the aim to obtain the \(L^2\)-index theorem, in the case \(M\) compact, as main result: “Suppose that \(\widetilde M\downarrow M\) is a normal covering of \(M\) with covering group \(\Gamma= \pi_1(M)/ \pi_1 (\widetilde M)\), \(P\) is an elliptic boundary problem on \(M\) and \(\widetilde P\) its lift to \(\widetilde M\). Then the numbers \(\dim_\Gamma \ker (\widetilde P)\), \(\dim_\Gamma \text{coker} (\widetilde P)\) are finite and \(\text{ind}_\Gamma (\widetilde P) (=\dim_\Gamma \ker (\widetilde P)- \dim_\Gamma \text{coker} (\widetilde P))= \text{ind} (P)\). Particularly, \(\text{ind}_\Gamma (\widetilde P)\) is an integer.”
This theorem is applied to the construction of geometric representations of Lie groups.